Electromagnetic Induction And Inductance

(a) Identify the circuit element ‘X’ in the circuit:

The circuit element ‘X’ where the current leads the voltage by is the capacitor. In capacitive circuits, the charging current for the capacitor leads the voltage across the capacitor.

(b) Write the formula for its reactance:

The reactance of a capacitor is given by:

where is the angular frequency ( , with being the frequency of the AC supply), and is the capacitance in farads.

(c) Show graphically the variation of this reactance with frequency of AC voltage:

The capacitive reactance , for a capacitor in an AC circuit is defined by the formula:

where represents the angular frequency, and is the capacitance. This formula shows that is inversely proportional to the frequency .

Graphical Representation:

To illustrate the relationship graphically:

1. Axes Setup: Plot frequency ( ) on the horizontal axis and capacitive reactance ( ) on the vertical axis.
2. Curve Behavior: The graph will display a hyperbolic decay as increases. At very low frequencies, the reactance is very high, representing nearly an open circuit. As increases, decreases, suggesting that the capacitor offers less impedance to higher frequency AC signals.
3. Key Points:
   • At , approaches infinity, which can be represented by the curve starting from the top of the y-axis.
   • As increases, decreases sharply initially and then more gradually, asymptotically approaching zero but never actually reaching it.

(d) Explain the behaviour of this element when it is used in:

(i) an AC circuit:

In an AC circuit, a capacitor impedes the flow of the current depending on the frequency of the AC supply. Higher frequencies reduce the reactance, allowing more current to pass through, demonstrating a characteristic called capacitive reactance.

(ii) a DC circuit:

In a DC circuit, a capacitor initially conducts as it charges, but once fully charged, it acts as an open circuit. Thus, after the initial charge period, no current flows through the capacitor in a steady-state DC circuit.